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This paper studies a class of minimax (saddle‑point) problems of the form max_{c in C} min_{β in S} g(c;β). The authors focus on the special but common case when the outcome is bilinear: g(c;β)=c^T A β, where C and S are compact convex sets. They show that for this matrix‑game structure the difficult step inside the Alternating Direction Method of Multipliers (ADMM) — the so‑called proximal operator — simplifies exactly to a generalized projection onto the confidence region S. No approximation or linearization is required.
At a high level, ADMM solves the problem by splitting it into easier subproblems and iterating. In general one substep requires solving a proximal minimax problem which can be as hard as the original problem. When g is bilinear the authors exchange the min and max and carry out the inner minimization in closed form. That algebra reduces the proximal step to solving a convex quadratic program: find β in S that minimizes ||Aβ−w||^2 for an explicit vector w. The remaining ADMM updates are then simple: form y = v + (1/ρ)Aβ, project y+u onto C (a standard Euclidean projection), and update the dual variable u.
The paper gives a concrete algorithm (Algorithm 1). Each iteration does a generalized projection onto S (line 4), an auxiliary update y (line 5), a Euclidean projection onto C (line 6), and a dual update (line 7). The generalized projection is a convex quadratic minimization over S. When A is the identity map this reduces to the familiar Euclidean projection. The algorithm ends by recovering the worst‑case parameter β* = argmin_{β∈S} g(c_k;β) for the final decision c_k.
Why this matters: many practical decisions are made under parameter uncertainty. The paper motivates the method with a marketing example. If β encodes conversion rates from randomized lift studies and c is a budget allocation, then the linearized incremental outcome can be written as c^T A β. A confidence set S for β (for example an ellipsoid or a likelihood‑ratio region for binomial counts) expresses uncertainty. Solving the resulting maximin problem finds a robust decision that performs best in the worst‑case β in S. The reduction shown here makes such robust solutions tractable for general convex confidence regions S.
Important caveats and limits are noted. The reduction to a generalized projection requires the bilinear form g(c;β)=c^T A β. For more general concave‑convex g the proximal step remains hard. The generalized projection itself is a convex quadratic program; that is easier than the original saddle problem but still needs a solver when S has a complicated shape. The method assumes C and S are compact and convex and uses a penalty parameter ρ and tolerances; convergence depends on these standard ADMM settings.
On convergence the authors invoke classical two‑block ADMM theory. Under the usual convexity and feasibility assumptions the iterates satisfy primal feasibility (consensus violation goes to zero), dual feasibility, and objective convergence to the optimum. The paper notes the standard worst‑case O(1/k) rate for the objective, while observing that ADMM often converges faster in practice. It also contrasts ADMM with other approaches: projected subgradient methods are slow, and accelerated proximal gradient methods can fail when the inner minimizer over β is not unique — a case common for bilinear problems — whereas ADMM remains reliable in those scenarios.